Electronic Journal of Differential Equations, Vol. 2006(2006), No. 140, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE OF PERIODIC SOLUTION FOR PERTURBED
GENERALIZED LIÉNARD EQUATIONS
ISLAM BOUSSAADA, A. RAOUF CHOUIKHA
Abstract. Under conditions of Levinson-Smith type, we prove the existence
of a τ -periodic solution for the perturbed generalized Liénard equation
t
u00 + ϕ(u, u0 )u0 + ψ(u) = ω( , u, u0 )
τ
with periodic forcing term. Also we deduce sufficient condition for existence
of a periodic solution for the equation
u00 +
2s+1
X
k=0
t
k
pk (u)u0 = ω( , u, u0 ).
τ
Our method can be applied also to the equation
t
u00 + [u2 + (u + u0 )2 − 1]u0 + u = ω( , u, u0 ).
τ
The results obtained are illustrated with numerical examples.
1. Introduction
Consider Liénard equation
u00 + ϕ(u)u0 + ψ(u) = 0
2
d u
00
1
where u0 = du
dt , u = dt2 , ϕ and ψ are C . Studying the existence of periodic
solution of period τ0 has been purpose of many authors: Farkas [3] presents some
typical works on this subject, where the Poincaré-Bendixson theory plays a crucial
role. In general, a periodic perturbation of the Liénard equation does not possess
a periodic solution as described by Moser; see for example [1].
Let us consider the perturbed Liénard equation
t
u00 + ϕ(u)u0 + ψ(u) = ω( , u, u0 )
τ
where ω is a controllably periodic perturbation in the Farkas sense; i.e., it is periodic
with a period τ which can be choosen appropriately. The existence of a non trivial
periodic solution for (2) was studied by Chouikha [1]. Under very mild conditions
it is proved that to each small enough amplitude of the perturbation there belongs
a one parameter family of periods τ such that the perturbed system has a unique
periodic solution with this period.
2000 Mathematics Subject Classification. 34C25.
Key words and phrases. Perturbed systems; Liénard equation; periodic solution.
c 2006 Texas State University - San Marcos.
Submitted May 15, 2006. Published November 1, 2006.
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I. BOUSSAADA, A. R. CHOUIKHA
EJDE-2006/140
Let us consider now the following generalized Liénard equation, which is “a more
realistic assumption in modelling many real world phenomena” as stated in [3, page
105]
u00 + ϕ(u, u0 )u0 + ψ(u) = 0.
(1.1)
Where ϕ and ψ are C 1 and satisfy some assumptions that will be specified below.
The leading work of investigation for the existence of periodic solution of generalized
Liénard systems was established by Levinson-Smith [4]. Let us define conditions
CLS .
Definition. The functions ϕ and ψ satisfy the condition CLS if: xψ(x) > 0 for
|x| > 0,
Z x
ψ(s)ds = Ψ(x) and
lim Ψ(x) = +∞, ϕ(0, 0) < 0.
x→+∞
0
Moreover, there exist some numbers 0 < x0 < x1 and M > 0 such that:
ϕ(x, y) ≥ 0
for |x| ≥ x0 ,
ϕ(x, y) ≥ −M for |x| ≤ x0
Z x1
x1 > x0 ,
ϕ(x, y(x))dx ≥ 10M x0
x0
for every decreasing function y(x) > 0.
Proposition 1.1 (Levinson-Smith [4]). When the functions ϕ and ψ are of class
C 1 and satisfy condition CLS then the generalized Lienard equation (1.1) has at
least one non-constant τ0 -periodic solution.
A non trivial solution will be denoted u0 (t), and its period τ0 . This proposition
has many improvements (under weaker hypotheses) due to Zheng and Wax Ponzo;
see [3], among other authors.
This article is organized as follows: At first, we prove the existence of a periodic
solution for the perturbed generalized Liénard equation
t
(1.2)
u00 + ϕ(u, u0 )u0 + ψ(u) = ω( , u, u0 ),
τ
Where t, , τ ∈ R are such that |τ − τ0 | < τ1 < τ0 , || < 0 with 0 ∈ R sufficient
small and τ1 is a fixed real scalar. We will use the Farkas method which was effective
for perturbed Liénard equation. In the third section, we will propose a criteria for
the existence of periodic solution for
u00 +
2s+1
X
k=0
t
k
pk (u)u0 = ω( , u, u0 ),
τ
(1.3)
with s ∈ N and pk are C 1 functions, for all k ≤ 2s + 1. In the second part of
the section, using a result of De Castro [2] we will prove uniqueness of a periodic
solution for the equation
u00 + [u2 + (u + u0 )2 − 1]u0 + u = 0.
(1.4)
Sufficient condition for the existence of periodic solution to
t
u00 + [u2 + (u + u0 )2 − 1]u0 + u = ω( , u, u0 ).
(1.5)
τ
will be found. At the end of the paper, some phase plane examples are given in
order to illustrate the above results. In particular, we describe uniqueness of a
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EXISTENCE OF PERIODIC SOLUTION
3
solution for equation (1.4) and the existence of a solution of equation (1.5) for
ω( τt , u, u0 ) = (sin 2t) u0 .
2. Periodic solution of perturbed generalized Lienard equation
In this part of this paper we prove the existence of periodic solution of the
perturbed generalized Lienard equation (1.2) such that the unperturbed one (1.1)
has at least one periodic solution. The method of proof that we will employ was
described in [1, 3].
Consider the equation (1.1) We assume that ϕ and ψ are C 1 and satisfy CLS .
Then by Proposition 1.1 there exists at least a non trivial periodic solution denoted
u0 (t).
Let the least positive period of the solution u0 (t) be denoted by τ0 and U be an
open subset of R2 containing (0, 0). These notation will be used in the rest of the
paper.
Theorem 2.1. Let ϕ and ψ be C 1 and satisfy CLS . Suppose 1 is a simple characteristic multiplier of the variational system associated to (1.1). Then there are two
real functions τ, h defined on U ⊂ R2 and constants τ1 < τ0 such that the periodic
solution ν(t, α, a + h(, α), , τ (, α)) of the equation
t
u00 + ϕ(u, u0 )u0 + ψ(u) = ω( , u, u0 ),
τ
exists for (, α) ∈ U , |τ − τ0 | < τ1 , τ (0, 0) = τ0 and h(0, 0) = 0.
We point out that the characteristic multipliers are the eigenvalues of the characteristic matrix which is the fundamental matrix in the time τ0 .
Proof of Theorem 2.1. Following the method used in [3], we set x2 = u , x1 = du
dt =
u0 and note x = col(x1 , x2 ) = col(u0 , u). The plane equivalent system of (1.1) is
0
x1 = −ϕ(x2 , x1 )x1 − ψ(x2 )
0
x = f (x) ⇐⇒
(2.1)
x02 = x1
with
f (x) = col(−ϕ(x2 , x1 )x1 − ψ(x2 ), x1 ).
Then the system (2.1) has the periodic solution q(t) with period τ0 . We define
q(t) = col(u0 0 (t), u0 (t))
and therefore
q 0 (t) = col(−ϕ(u0 (t), u0 0 (t))u0 0 (t) − ψ(u0 (t)), u0 0 (t)).
The variational system associated with (2.1) is
y 0 = fx 0 (q(t))y,
(2.2)
Without loss of generality, we take the initial conditions
t = 0,
u0 (0) = a < 0
and u00 (0) = 0
Hence fx 0 (q(t)) is the matrix
0
−ϕ x1 (u0 (t), u0 0 (t))u0 0 (t) − ϕ(u0 (t), u0 0 (t)) −ϕ0 x2 (u0 (t), u0 0 (t))u0 0 (t) − ψ 0 (u0 (t))
1
0
Notice that q 0 (t) = col(−ϕ(u0 (t), u0 0 (t))u0 0 (t) − ψ(u0 (t), u0 0 (t)) is the first solution
of the variational system. Now we calculate the second one, denoted by yb(t) =
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I. BOUSSAADA, A. R. CHOUIKHA
EJDE-2006/140
col(b
y1 (t), yb2 (t)) and linearly independent with q 0 (t) = y(t), in order to write the
fundamental matrix. Consider
h Z s
i
I(s) = exp −
(ϕ0 x1 (u0 (ρ), u0 0 (ρ))u0 0 (ρ) + ϕ(u0 (ρ), u0 0 (ρ)))dρ
0
and
Z
t
−2 0
ϕ(u0 (ρ), u0 0 (ρ))u0 0 (ρ) + ψ(u0 (ρ))
ϕ x2 (u0 (t), u0 0 (t))u0 0 (t)
0
+ ψ 0 (u0 (t)) I(ρ)dρ
π(t) = −
We then obtain
yb1 (t) = −[ϕ(u0 (t), u0 0 (t))u0 0 (t) + ψ(u0 (t)]π(t),
yb2 (t) = u0 0 (t)π(t) + π 0 (t)
ϕ(u0 (t), u0 0 (t))u0 0 (t) + ψ(u0 (t)
ϕ0 x2 (u0 (t), u0 0 (t))u0 0 (t) + ψ 0 (u0 (t))
It is known, [1, 3], that the fundamental matrix satisfying Φ(0) = Id2 is Φ(t) equals
to
ϕ(u0 (t),u0 0 (t))u0 0 (t)+ψ(u0 (t))
ψ(a)
0
0 (t)
− uψ(a)
ψ(a)π(t)[ϕ(u0 (t), u0 0 (t))u0 0 (t) + ψ(u0 (t)]
0
0
0 (t))u0 (t)+ψ(u0 (t))
−ψ(a)u00 (t)π(t) − ψ(a)π 0 (t) ϕ0ϕ(u(u0 (t),u
0
0
0
0 (t),u0 (t))u0 (t)+ψ (u0 (t))
x2
Thus,
Φ(τ0 ) =
2
1 ψ(a) π(τ0 )
.
0
ρ2
We use the Liouville’s formula
Z
det Φ(t) = det Φ(0) exp
t
Tr(fx 0 (q(τ )))dτ.
0
Since det(Φ(0)) = 1, we deduce
h Rthe characteristic multipliers associated with (2.2):
i
τ
ρ1 = 1 and ρ2 = I(τ0 ) = exp − 0 0 (ϕ0 x1 (u0 (ρ), u0 0 (ρ))u0 0 (ρ)+ϕ(u0 (ρ), u0 0 (ρ)))dρ .
From [3], we have:
−ψ(a) 0
J(τ0 ) = −Id2 +
+ Φ(τ0 )
0
0
Hence we obtain the jacobian matrix
2
−ψ(a) ψ(a) π(τ0 )
J(τ0 ) =
,
0
ρ2 − 1
Since 1 is a simple characteristic multiplier (ρ2 6= 1), det J(0, 0, 0, τ0 ) 6= 0. We
define the periodicity condition
z(α, h, , τ ) := ν(α + τ, a + h, , τ ) − (a + h) = 0.
(2.3)
By the Implicit Function Theorem there are 0 > 0 and α0 > 0 and uniquely
determined functions τ and h defined on U = {(α, ) ∈ R2 : || < 0 , |α| < α0 } such
that: τ, h ∈ C 1 , τ (0, 0) = T0 , h(0, 0) = 0 and z(α, h, , τ ) ≡ 0. Because of (2.3), the
periodic solution of (1.2) has period τ (, α) near T0 and has path near the path of
the unperturbed solution.
!
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EXISTENCE OF PERIODIC SOLUTION
In particular if ρ2 < 1, the periodic solution is orbitally asymptotically
i.e. stable in the Liapunov sense and it is attractive see [3, page 346].
the following inequality is a criteria of the existence of orbital asymptotical
periodic solution of the equation (1.2).
Z τ0
(ϕ0 x1 (u0 (ρ), u0 0 (ρ))u0 0 (ρ) + ϕ(u0 (ρ), u0 0 (ρ)))dρ > 0.
ρ2 < 1 ⇐⇒
5
stable
Thus,
stable
(2.4)
0
Using Proposition 1.1, we conclude the existence of non trivial periodic solution for
perturbed generalized Liénard equation.
3. Results on the periodic solutions
Special case. Let us now consider the equation
u00 +
2s+1
X
k
pk (u)u0 = 0.
(3.1)
k=0
Let pk be C 1 function, for allk ≤ 2s + 1 for s ∈ N. This is a special case of Liénard
equation with p0 (u) = ψ(u) and
ϕ(u, u0 ) =
2s+1
X
pk (u)u0
k−1
.
k=1
We will suppose ϕ and ψ verify CLS conditions. Let U be an open subset of R2
containing (0, 0). The associated perturbed equation, as denoted previously (1.3),
is equation
2s+1
X
t
k
pk (u)u0 = ω( , u, u0 ).
u00 +
τ
k=0
P2s+1
0k
Remark. The last non-zero term of the finite sum
has an odd
k=0 pk (u)u
index. Then it is necessary to have the element x0 6= 0 in the CLS conditions.
Theorem 3.1. Let ϕ and ψ be C 1 and satisfy CLS . If 1 is a simple characteristic
multiplier of the variational system associated to (3.1) then there are two functions
τ, h : U → R and constants τ1 < τ0 such that the periodic solution ν(t, α, a +
h(, α), , τ (, α)) of the equation
u00 +
n
X
k=0
t
k
pk (u)u0 = ω( , u, u0 )
τ
exists for (, α) ∈ U with |τ − τ0 | < τ1 , τ (0, 0) = τ0 and h(0, 0) = 0.
Proof. We will use the same method as in the existence theorem for non-trivial
periodic solution of the perturbed system. Consider the unperturbed equation
to compute some useful elements. First we assume that 2s + 1 = n, to simplify
0
notation. Let x2 = u and x1 = du
dt = u . The equivalent plane system of (3.1) is
0
Pn
x1 = − k=0 pk (x2 )x1 k
0
x = f (x) ⇐⇒
(3.2)
x02 = x1
with
f (x) = col(−
n
X
k=0
pk (x2 )x1 k , x1 ).
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I. BOUSSAADA, A. R. CHOUIKHA
EJDE-2006/140
Let q(t) = col(u00 (t), u0 (t)) the periodic solution of (3.2). The variational system
associated to (3.2) is
y 0 = fx0 (q(t))y
with the periodic solution
0
q (t) = col(−
n
X
k
pk (u0 )(t)u00 (t), u00 (t)),
k=0
hence
fx0 (q(t))
Pn
k−1
− k=1 kpk (u0 (t))u00 (t)
=
1
−
k
Pn
k=0
p0k (u0 (t))u00 (t)
0
.
We assume the initial values:
t = 0,
u0 (0) = a < 0
and u0 0 (0) = 0.
Then q(0) = col(0, a) and q 0 (0) = col(−ψ(a), 0).
In same way as the previous section we compute the fundamental matrix associated with (3.2), denoted Φ(t). Determine the second vector solution (linearly
independent with q 0 (t) = y(t)). A trivial calculation described in [1, 3] gives us the
y(t)
, y(0)b
y (t)). For that consider
second solution denoted yb(t), hence Φ(t) = ( y(0)
h
s
Z
I(s) = exp −
(
0
n
X
k−1
kpk (u0 (ρ))u00 (ρ)
i
)dρ ,
k=1
and denote as in the previous section
Z
π(t) = −
t
(
0
n
X
pk (u0 )(ρ)u00 (ρ)k )−2 (
k=0
n
X
k
p0k (u(t))u0 (t))I(ρ)dρ.
k=0
Sine yb(t) = col(b
y1 (t), yb2 (t)), where
yb1 (t) = −(
n
X
pk (u0 )(t)u00 (t)k )π(t),
k=0
Pn
k
pk (u0 )(t)u00 (t)
yb2 (t) = u00 (t)π(t) + π 0 (t) Pnk=0
.
k
0
0
k=0 pk (u0 (t))u0 (t)
Hence the fundamental matrix associated with our variational system is

 Pn
0k
Pn
0
k
k=0 pk (u0 )(t)u0 (t)
p
(u
)(t)u
(t)
)π(t)
ψ(a)(
k
0
0
k=0
ψ(a)
.
Pn
Φ(t) = 
0
p (u )(t)u0 (t)k
0 (t)
− uψ(a)
−ψ(a)u00 (t)π(t) − ψ(a)π 0 (t) Pnk=0 p0k (u0 (t))u00 (t)k
k=0
k
We deduce the principal matrix (the fundamental one with t = τ0 ).
2
1 ψ(a) π(τ0 )
Φ(τ0 ) =
.
0
ρ2
0
0
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EXISTENCE OF PERIODIC SOLUTION
7
By the Liouville’s formula, we have the characteristic multipliers ρ1 = 1 and
ρ2 = det(Φ(τ0 ))
Z τ0
= exp
(T rfx 0 (q(τ ))dτ
0
τ0
Z
= exp −
0
n
X
k−1
kpk (u0 (τ ))u00 (τ )
)dτ
k=1
Then we define the equivalence (2.4):
Z τ0 X
n
k−1
ρ2 < 1 ⇐⇒
kpk (u0 (τ ))u00 (τ )
dτ > 0
0
(3.3)
k=1
and the associated Jacobian matrix is
2
−ψ(a) ψ(a) π(τ0 )
J(τ0 ) =
.
0
ρ2 − 1
Uniqueness of the periodic solution for an unperturbed equation. Let us
consider now equation (1.4):
u00 + [u2 + (u + u0 )2 − 1]u0 + u = 0,
which is a special case of generalized Liénard equation with
ϕ(u, u0 ) = (u2 + (u0 + u)2 − 1) and ψ(u) = u.
We will prove existence and uniqueness of non trivial periodic solution for equation
(1.4). Existence will be ensured by CLS conditions and for proving uniqueness we
use a De Castro’s result [5] (see also [2]).
Proposition 3.2 (De Castro [1]). Suppose the following system has at least one
periodic orbit
y 0 = −ϕ(x, y)y − ψ(x)
x0 = y.
Then under the following two assumptions:
(a) ψ(x) = x;
(b) ϕ(x, y) increases, when |x| or |y| or the both increase
this periodic orbit is unique.
Let us verify that (1.4) satisfies the above assumptions: Equation (1.4) is satisfied
if and only if
u00 +
3
X
k
pk (u)u0 = 0,
(3.4)
k=0
p0 (u) = ψ(u) = u,
2
p1 (u) = 2u − 1,
p2 (u) = 2u,
p3 (u) = 1.
Also if and only if
u00 + ϕ(u, u0 )u0 + ψ(u) = 0,
ϕ(u, u0 ) = (u2 + (u0 + u)2 − 1),
ψ(u) = u.
(3.5)
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I. BOUSSAADA, A. R. CHOUIKHA
EJDE-2006/140
Clearly, the assumptions of Proposition 3.2 are satisfied. In the following, we firstly
verify conditions CLS . In that case the equation
u00 + ϕ(u, u0 )u0 + ψ(u) = 0
has at least a non trivial periodic solution. It is easy to see that ψ(u) = u satisfies
xψ(x) > 0
for |x| > 0,
x
Z
ψ(s)ds = Ψ(x),
limx→+∞ Ψ(x) = +∞
0
Now we have ϕ(0, 0) = −1 < 0. By taking x0 = 1, M = 1, we have
ϕ(x, y) ≥ 0
ϕ(x, y) ≥ −M
for |x| ≥ x0 ,
for |x| ≤ x0 .
The following calculation gives us the optimal value of x1 > x0 . Let
Z x1
H=
ϕ(x, y)dx
x0
Z x1
=
[x2 + (x + y)2 − 1]dx
1
Z x1
=
[2x2 + 2xy + y 2 − 1]dx
1
=
h2
3
ix1
x3 + x2 y + x(y 2 − 1)
1
x1 2 − 2x1 + 1
x1 + 1 2
x1 + 1
= (x1 − 1)(
+ 2(
) + 2y(
) + (y 2 − 1))
6
2
2
x 2 − 2x + 1
x1 + 1 1
1
+ ϕ(
, y)
= (x1 − 1)
6
2
Since
H>
x1 +1
> x0 =
2
(x1 −1)3
. Hence,
6
1, using the inequality ϕ(x, y) ≥ 0 for |x| ≥ x0 , we obtain
if
(x1 −1)3
6
1
= 10M x0 = 10, then x1 = 1 + (60) 3 which satisfies
Z x1
x1 > x0 ,
ϕ(x, y) dx ≥ 10M x0 ,
x0
for every decreasing function y(x) > 0.
Existence of periodic solution for perturbed equation satisfying CLS . In
the following we are dealing with the existence of periodic solution for the equation
(1.5). We assume the initial values:
t = 0,
u0 (0) = a < 0,
u0 0 (0) = 0.
Theorem 3.3. Suppose 1 is a simple characteristic multiplier of the variational
system associated to (1.4). Then there are two functions τ, h : U → R and constants
τ1 < τ0 such that the periodic solution ν(t, α, a + h(, α), , τ (, α)) of the equation
t
3
2
u00 + u0 + 2uu0 + (2u2 − 1)u0 + u = ω( , u, u0 ),
τ
exists for (, α) ∈ U with |τ − τ0 | < τ1 , τ (0, 0) = τ0 and h(0, 0) = 0.
EJDE-2006/140
EXISTENCE OF PERIODIC SOLUTION
9
Proof. We proceed similarly as in the proof of Theorem 3.1. We substitute the
fundamental matrix, the second characteristic multiplier is ρ2 . The following holds
for equation (1.4),
Z τ0 X
3
k−1
ρ2 < 1 ⇐⇒
(
kpk (u0 (τ ))u00 (τ )
)dτ > 0,
0
then
Z
ρ2 < 1 ⇐⇒
τ0
k=1
2
[2u0 2 (τ ) + 4u0 (τ )u00 (τ ) + 3u00 (τ ) − 1]dτ > 0.
0
It ensures that 1 is a simple characteristic multiplier of the variational system
associated to (1.4) it implies J(τ0 ) 6= 0. Then a periodic solution for the perturbed
equation (1.5) exists.
Using Scilab we will describe the phase plane of equation (1.4) u00 + [u2 + (u +
u ) − 1]u0 + u = 0. We take x0 = u0 (0) = a = −0.7548829, y0 = u0 0 (0) = 0 and the
step time of integration (step = .0001). Recall that the periodic orbit is unique.
0 2
Figure 1. (A) The unique periodic orbit for u00 +[u2 +(u+u0 )2 −
1]u0 + u = 0. (B) Zoom on the periodic orbit (×20)
We take ω( τt , u, u0 ) = sin(2t)u0 . Some illustrations of the phase portrait for
the perturbed equation (1.5), those can explain existence of a bound 0 , from which
periodicity of the orbit will be not insured. In order to localize 0 , we have taken
several values of .
Figure 2. (C) The periodic orbit for u00 +[u2 +(u+u0 )2 −1]u0 +u =
ω( τt , u, u0 ), = 0.001. (D) Zoom on the periodic orbit (×20)
10
I. BOUSSAADA, A. R. CHOUIKHA
EJDE-2006/140
Figure 3. (E) Orbit for u00 +[u2 +(u+u0 )2 −1]u0 +u = ω( τt , u, u0 ),
= 0.01. (F) Zoom on the orbit (×10) and loss of periodicity.
We see that from the range of = 0.01 the orbit loses the periodicity.
Table 1. Period τ for some values of τ
τ
0
1/1000 1/900 1/800 1/700
5.4296 5.4287 5.4286 5.4285 5.4283
1/600 1/500 1/400 1/300 1/200
5.4281 5.4278 5.4274 5.4267 5.4252
Acknowledgements. We thank Professors Miklos Farkas and Jean Marie Strelcyn
for their helpful discussions; also the referees for their suggestions.
References
[1] A. R. Chouikha, Periodic perturbation of non-conservative second order differential equations,
Electron. J. Qual. Theory. Differ. Equ, 49 (2002), 122-136.
[2] A. De Castro, Sull’esistenza ed unicita delle soluzioni periodiche dell’equazione ẍ + f (x, ẋ)ẋ +
g(x) = 0, Boll. Un. Mat. Ital, (3) 9 (1954). 369–372.
[3] M. Farkas, Periodic motions, Springer-Verlag, (1994).
[4] N. Levinson and O. K. Smith, General equation for relaxation oscillations, Duke. Math. Journal., No 9 (1942), 382-403.
[5] R. Reissig G. Sansonne R. Conti; Qualitative theorie nichtlinearer differentialgleichungen,
Publicazioni delĺ instituto di alta matematica, (1963).
Islam Boussaada
LMRS, UMR 6085, Universite de Rouen, Avenue de l’université, BP.12, 76801 Saint Etienne du Rouvray, France
E-mail address: islam.boussaada@etu.univ-rouen.fr
A. Raouf Chouikha
Universite Paris 13 LAGA, Villetaneuse 93430, France
E-mail address: chouikha@math.univ-paris13.fr